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The Laplace transform is $\hat{f}(p)={\int }_0^{\infty }{e}^{-pt}f(t),dt$, and inversion iswhere the vertical Bromwich contour lies to the right of all singularities. For $t>0$ it is closed leftwards, and for $t<0$ rightwards, where the exponential decays.

Transforming the PDE and imposing boundedness as $x\to \infty$ giveswith the square root cut conventionally along $(-\infty ,0]$. ThereforeThere are poles at $p=0,-1$ (with cancellation assessed in the complete expression) and a branch point at $0$; close left for $t>0$ and wrap the cut while including the relevant residues.

Logistic regression assumes independent $Y_i\mid X_i\sim \mathrm{Bernoulli}({\pi }_i)$ withFor the binomial exponential family, $\ell =y\theta -m\log (1+e^{\theta })+\text{constant}$, so the mean is $m{e}^{\theta }/(1+e^{\theta })$ and the canonical parameter is its logit. It may be fitted by iteratively reweighted least squares.

Under LDA, Bayes' rule giveswhich is a logistic model with an intercept.